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Claim. Under the continuum hypothesis there is a family $(A_x)_{x\in\mathbb R}$ of bounded sets of outer measure less than $1$, in fact countable and null, with no infinite independent set, so the first question of Problem 501 has a negative answer under CH. The construction, as the problem page writes it: enumerate $\mathbb R=\{r_\alpha:\alpha<\omega_1\}$ and put $A_{r_\beta}=\{r_\alpha:\alpha<\beta,\ |r_\alpha|\le|r_\beta|+1\}$; each set is countable and bounded by $|r_\beta|+1$, and an infinite independent set would give $|x_0|>|x_1|+1>|x_2|+2>\cdots$, which is impossible. The site attributes this result to Hechler [He72]. The site's curator, in a comment of 31 August 2025 on the problem's discussion thread, wrote that this paper addresses the first question and shows it false under CH, besides a separate theorem on part (A) of the Erdős–Hajnal problem. The paper itself has not been read for this page, and zbMATH carries no review of it; the construction is written out in Lee's and Glazer's notes.

Hypothesis. The continuum hypothesis, which ZFC neither proves nor refutes. The claim settles neither question alone; with a positive answer in another model of ZFC it gives the independence of the first question, as on Glazer's claim page and, relative to a measurable cardinal, on Lee's claim page.

Covers. The first question of Problem 501 (the part first_question), on its not-provable side: the family above exists in every model of ZFC with the continuum hypothesis, and that hypothesis holds in Gödel's constructible universe, so it is consistent with ZFC if ZFC is consistent; ZFC therefore does not prove that every family of bounded sets of outer measure below one has an infinite independent set. That side alone leaves the question open. Together with Glazer's claim page, which settles the not-disprovable side relative to the consistency of ZFC alone, it shows that the first question is independent of ZFC if ZFC is consistent; Glazer's page carries both sides, its negative half being this counterexample. It does not cover the second question.

Source. S. H. Hechler, On two problems in combinatorial set theory, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 429–431 (zbMATH 0252.04002). The issue month is not known, so this page is dated to the first day of 1972. P. Erdős and A. Hajnal, Unsolved and solved problems in set theory, Proc. Sympos. Pure Math. 25 (1974), 269–287, credit Hechler on p. 279, under Problem 38, with a proof that under MA their Problem 38/C, the first question, fails even when every $A_x$ is null; their MA (p. 269) is $\mathrm{MA}_\kappa$ for every $\kappa<2^{\aleph_0}$, which CH implies. Which Hechler paper contains the counterexample is not settled by the sources: Erdős and Hajnal cite a Hechler preprint, A note on some topological problems of Erdős (their reference [18], p. 287); Komjáth's survey lists [He72] under Problem 38(A) of the Erdős–Hajnal list; the formal-conjectures statement file ErdosProblems/501.lean cites as [He72] Hechler's A dozen small uncountable cardinals (TOPO 72, Lecture Notes in Math., 1972); and the Lean development of Glazer's claim cites Hechler's paper in Israel J. Math. 11 (1972), 231–248.

Acceptance. Reviewed: the curator of erdosproblems.com, T. F. Bloom, credits [He72] with the negative answer under CH in the commentary of a problem the site labels settled (NOT DISPROVABLE), and is independent of Hechler. Not refereed as recorded here: the theorem has not been read in the paper or in a review of it.

Depends on. No other wiki page; the claim rests on the cited paper as the site credits it.